Contact seaweeds

Coll, Vincent E.; Mayers, Nicholas; Russoniello, Nicholas; Salgado, G.

doi:10.2140/pjm.2022.320.45

Cited by 4 publications

(1 citation statement)

References 11 publications

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“…The first two papers of this sequence establish Theorem 6 for type-A (Coll et al [10], 2022) and type-C (Coll and Russoniello [11], 2022) seaweeds by showing, in these classical cases, that index one is sufficient to identify that the seaweed is contact. This is unexpected, as a contact Lie algebra has index one -but not characteristically so.…”

Section: Introductionmentioning

confidence: 99%

Classification of contact seaweeds

Coll¹,

Russoniello²

2023

Preprint

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A celebrated result of Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. In general, such structures are not invariant under left translation. The problem of finding which Lie groups admit a left-invariant contact structure (contact Lie groups) resolves to the question of determining when a Lie algebra g is contact; that is, admits a one-form ϕ ∈ g * such that ϕ ∧ (dϕ) k = 0.In full generality, this remains an open question; however we settle it for the important category of the evocatively named seaweed algebras by showing that an index-one seaweed is contact precisely when it is quasi-reductive. Seaweeds were introduced by Dergachev and Kirillov who initiated the development of their index theory -since completed by Joseph, Panyushev, Yakimova, and Coll, among others. Recall that a contact Lie algebra has index one -but not characteristically so. Leveraging recent work of Duflo, Khalgui, Torasso, Panyushev, Yakimova, and Ammari, who collectively classified quasi-reductive seaweeds, our equivalence yields a full classification of contact seaweeds. We remark that since type-A and type-C seaweeds are de facto quasi-reductive (by a result of Panyushev), in these types index one alone suffices to ensure the existence of a contact form.

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Section: Introductionmentioning

confidence: 99%